cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A378238 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3*n+r+k,n)/(3*n+r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 14, 0, 1, 6, 32, 134, 0, 1, 8, 54, 324, 1482, 0, 1, 10, 80, 578, 3696, 17818, 0, 1, 12, 110, 904, 6810, 45316, 226214, 0, 1, 14, 144, 1310, 11008, 85278, 583152, 2984206, 0, 1, 16, 182, 1804, 16490, 140936, 1113854, 7769348, 40503890, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2024

Keywords

Examples

			Square array begins:
  1,      1,      1,       1,       1,       1,       1, ...
  0,      2,      4,       6,       8,      10,      12, ...
  0,     14,     32,      54,      80,     110,     144, ...
  0,    134,    324,     578,     904,    1310,    1804, ...
  0,   1482,   3696,    6810,   11008,   16490,   23472, ...
  0,  17818,  45316,   85278,  140936,  216002,  314700, ...
  0, 226214, 583152, 1113854, 1870352, 2914790, 4320608, ...

Crossrefs

Columns k=0..3 give A000007, A144097, A371675, A365843.
T(n,n) gives 1/4 * A370102(n) for n > 0.
Cf. A033877, A071949, A378236, A378237, A378239, A378240.

Programs

  • PARI
    T(n, k, t=3, u=1) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + A_k(x)^(1/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A144097.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+3) for n > 0.

A371693 G.f. satisfies A(x) = ( 1 + x * A(x) * (1 + A(x)) )^2.

Original entry on oeis.org

1, 4, 28, 248, 2480, 26688, 301648, 3531424, 42449088, 520858496, 6497190528, 82146802944, 1050370074624, 13559126110720, 176469550681344, 2313050095245824, 30506619439926272, 404558181197010944, 5391161355764205568, 72156618656648237056, 969557980700415827968
Offset: 0

Views

Author

Seiichi Manyama, Apr 03 2024

Keywords

Crossrefs

Column k=2 of A378239.
Cf. A006318, A371681.
Cf. A219534.

Programs

  • PARI
    a(n, r=2, t=2, u=2) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

G.f.: B(x)^2 where B(x) is the g.f. of A219534.
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(2*n+2*k+2,n)/(n+k+1).

A378237 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+3*r+k,n)/(n+3*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 10, 0, 1, 6, 24, 74, 0, 1, 8, 42, 188, 642, 0, 1, 10, 64, 350, 1680, 6082, 0, 1, 12, 90, 568, 3234, 16212, 60970, 0, 1, 14, 120, 850, 5440, 31878, 164584, 635818, 0, 1, 16, 154, 1204, 8450, 54888, 328426, 1732172, 6826690, 0, 1, 18, 192, 1638, 12432, 87402, 574848, 3494142, 18728352, 74958914, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2024

Keywords

Examples

			Square array begins:
   1,     1,      1,      1,      1,      1,       1, ...
   0,     2,      4,      6,      8,     10,      12, ...
   0,    10,     24,     42,     64,     90,     120, ...
   0,    74,    188,    350,    568,    850,    1204, ...
   0,   642,   1680,   3234,   5440,   8450,   12432, ...
   0,  6082,  16212,  31878,  54888,  87402,  131964, ...
   0, 60970, 164584, 328426, 574848, 931770, 1433544, ...

Crossrefs

Columns k=0..1 give A000007, A349310.
Cf. A033877, A071949, A378236, A378238, A378239, A378240.

Programs

  • PARI
    T(n, k, t=1, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(3/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A349310.
B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k+3) for n > 0.

A378236 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+2*r+k,n)/(n+2*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 8, 0, 1, 6, 20, 44, 0, 1, 8, 36, 120, 280, 0, 1, 10, 56, 236, 800, 1936, 0, 1, 12, 80, 400, 1656, 5696, 14128, 0, 1, 14, 108, 620, 2960, 12192, 42416, 107088, 0, 1, 16, 140, 904, 4840, 22592, 92960, 326304, 834912, 0, 1, 18, 176, 1260, 7440, 38352, 176800, 727824, 2572992, 6652608, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2024

Keywords

Examples

			Square array begins:
   1,     1,     1,     1,      1,      1,      1, ...
   0,     2,     4,     6,      8,     10,     12, ...
   0,     8,    20,    36,     56,     80,    108, ...
   0,    44,   120,   236,    400,    620,    904, ...
   0,   280,   800,  1656,   2960,   4840,   7440, ...
   0,  1936,  5696, 12192,  22592,  38352,  61248, ...
   0, 14128, 42416, 92960, 176800, 308560, 507152, ...

Crossrefs

Columns k=0..1 give A000007, A346626.
Cf. A033877, A071949, A378237, A378238, A378239, A378240.

Programs

  • PARI
    T(n, k, t=1, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(2/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A346626.
B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k+2) for n > 0.

A378240 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3*n+3*r+k,n)/(3*n+3*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 18, 0, 1, 6, 40, 234, 0, 1, 8, 66, 540, 3570, 0, 1, 10, 96, 926, 8400, 59586, 0, 1, 12, 130, 1400, 14706, 141876, 1053570, 0, 1, 14, 168, 1970, 22720, 251622, 2528760, 19392490, 0, 1, 16, 210, 2644, 32690, 394152, 4524786, 46815116, 367677090, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2024

Keywords

Examples

			Square array begins:
  1,       1,       1,       1,       1,        1,        1, ...
  0,       2,       4,       6,       8,       10,       12, ...
  0,      18,      40,      66,      96,      130,      168, ...
  0,     234,     540,     926,    1400,     1970,     2644, ...
  0,    3570,    8400,   14706,   22720,    32690,    44880, ...
  0,   59586,  141876,  251622,  394152,   575402,   801948, ...
  0, 1053570, 2528760, 4524786, 7156128, 10553970, 14867704, ...

Crossrefs

Columns k=0..1 give A000007, A364167.
Cf. A033877, A071949, A378236, A378237, A378238, A378239.

Programs

  • PARI
    T(n, k, t=3, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + A_k(x)^(3/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A364167.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+5). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+5) for n > 0.
Showing 1-5 of 5 results.

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